Cloud Function & client authentication during HTTPS Call using firebase admin Auth

Background: I have three entities (language: JS):

1. Client on Browser
2. Google Cloud Functions
3. An HTTP Server [Have Firebase admin Auth middleware]

So, From 1 and 2, I want to communicate with 3 using HTTPS calls. Because the 3 has admin Auth already, I am able to authorize the calls from clients(1).

But I want to know how I can authorize calls from google cloud functions(2) so that I can securely communicate with both entities.

Token Auth Middleware:

const validateFirebaseIdToken = async (req: any, res: any, next: any) => {
  console.log(await auth.getCredentials());
  if (
    (!req.headers.authorization ||
      !req.headers.authorization.startsWith("Bearer ")) &&
    !(req.cookies && req.cookies.__session)
  ) {
    console.log(
      "No Firebase ID token was passed as a Bearer token in the Authorization header." +
        "Make sure you authorize your request by providing the following HTTP header:" +
        "Authorization: Bearer <Firebase ID Token>" +
        'or by passing a "__session" cookie.'
    );
    res.status(403).send("Unauthorized 1");
    return;
  }

  let idToken;
  if (
    req.headers.authorization &&
    req.headers.authorization.startsWith("Bearer ")
  ) {
    console.log('Found "Authorization" header');
    // Read the ID Token from the Authorization header.
    idToken = req.headers.authorization.split("Bearer ")[1];
  } else if (req.cookies) {
    console.log('Found "__session" cookie');
    // Read the ID Token from cookie.
    idToken = req.cookies.__session;
  } else {
    // No cookie
    res.status(403).send("Unauthorized 2");
    return;
  }

  try {
    const decodedIdToken = await admin.auth().verifyIdToken(idToken);
    req.user = decodedIdToken;
    next();
    return;
  } catch (error) {
    console.log("Error while verifying Firebase ID token:" + error);
    res.status(403).send("Unauthorized 3");
    return;
  }
};